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All issues Volume 8 (2024) Acta Acust., 8 (2024) 34 Full HTML
Open Access
Issue
Acta Acust.
Volume 8, 2024
Article Number 34
Number of page(s) 16
Section Virtual Acoustics
DOI https://doi.org/10.1051/aacus/2024023
Published online 09 September 2024

© The Author(s), Published by EDP Sciences, 2024

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

The concept of a diffuse sound field refers to uncorrelated unit-variance plane waves impinging from all directions. In an ‘ideal’, cylindrically or spherically isotropic diffuse field, the sound pressure level is position-independent and the time-averaged sound intensity vector vanishes at all positions [1]. Measurement of sound field diffuseness has an extensive history in acoustics, for instance based on the well-defined frequency-dependent correlation between two points in space [2–5]. In addition, the directional distribution of the incident sound [6] or the sound intensity vector can be used to quantify the diffuseness of the sound field [7–12]. In particular, sound intensity normalized by the energy density and speed of sound were defined as energy vector [13, 14] when the diffuse sound field is synthesized by incoherently driven loudspeakers, as it simplifies to the gain-square-weighted average of the loudspeaker directions. The resulting vector is of unit length for fully directional sound fields and zero length for fully diffuse sound fields. More recent approaches employed spherical microphone arrays to measure the diffuseness of the sound field, e.g. COMEDIE, which is based on an eigenvalue decomposition of the spherical harmonics (SH) covariance matrix [15]. A directional and temporal histogram of the energy decay in a simulated room was proposed in auralization [16, 17]. Several recent works moreover showed ways to retrieve or represent anisotropic reverberation properties based on measurements [18–30].

Synthesis of diffuse sound fields with loudspeaker arrays is of interest in laboratory environments, for example to quantify the diffuse-field sound transmission of building partitions [31] or acoustic absorption [32, 33], the diffuse-field response of microphone arrays [34–36] and dummy heads [37–39], or to measure the performance and perceptual quality of active noise control/cancellation algorithms [40–42], etc.

Spatial sound reproduction targets a faithful reproduction of diffuse sound fields that perceptually elicits a sensation of envelopment for the audience, by suitably employing surrounding loudspeakers [43–47].

Theoretically, sound field synthesis techniques such as wave field synthesis (WFS) [48–55] or near-field compensated higher-order Ambisonics (NFC-HOA) [52, 55–59] should allow to reproduce an ideally diffuse sound field, given a continuous distribution of elementary sources. In discrete-direction implementations, however, spatial aliasing severely affects the sound field synthesis capacities of both WFS and HOA in much of the auditory range. Nevertheless, it could be argued that the fruitfulness of these approaches lies in much more tolerant psychoacoustic effects of auditory localization [60], so that playback qualities allow for a much larger sweet area, as analyzed in experiments and perceptual measures [61–65].

In WFS, it was studied for diffuse-field synthesis how many uncorrelated plane-wave sources are needed for a centered listener [66, 67]. In Ambisonics, the required directional resolution or Ambisonic order for a large sweet area was investigated [68], and a directionally non-uniform reverberation algorithm was suggested [69].

The minimum number of loudspeakers and horizontal arrangements for reproducing the spatial impression of diffuse sound fields in channel-based playback was investigated in terms of interaural cross correlation [70] and listening experiments [44] with uncorrelated signals.

Interaural coherence and interaural level differences were moreover considered to evaluate the quality of diffuse-field reproduction capabilities [43, 46, 71].

Adding height loudspeaker layers slightly increases perceived diffuseness, when diffuseness is defined as ‘the sound coming from all directions with equal intensity. Therefore, the sound should ideally be impossible to localise and without any gaps [...] in three dimensions’ as shown in [45, 72]. Other researchers showed that two distinct spatial attributes related to the perception of diffuse sound fields can be separated, namely ‘envelopment’ for being surrounded by sound and ‘engulfment’ for being covered by sound [47, 73, 74]. Isotropy as technical measure of directional uniformity was recently discussed [22], and spherical t-designs were shown to be optimal plane-wave layouts to synthesize isotropic sound fields [75].

Reverberant spatial audio objects and diffuse-field modeling were recently proposed [26, 27] to model the typical/anisotropic characteristics of acoustic rooms. Bandpass-based decorrelation for Diffuse Field Modeling that improves diffuseness in measured 1st-order Ambisonic room impulse responses was proposed. By manipulation of the directional weights for front, left, and right, the perception of envelopment was shown to be affected to some degree by direction dependent diffuse fields [26]. Further investigating the perceivability of anisotropy in reverberation, i.e., by recognizing a rotation of the scene, revealed that a time averaged interaural energy decay difference at 850 Hz of slightly more than 1 dB can already be heard [25]. Simplified auralization of non-uniform room reverberation was moreover proposed and tested [76], using a warped, sparse layout of virtual reverberation sources.

In loudspeaker-based reproduction [77], channel-based, scene-based (Ambisonics), and object-based amplitude panning concepts mostly employ point-source loudspeakers for playback, yielding a sound pressure decay with distance and parallax with regard to the listening position. Object-based WFS loudspeaker systems are typically 2.5D and based on dense horizontal point-source loudspeaker arrays. When rendering plane-wave sound objects, they yield a sound pressure decay with distance, depending on the listening position, cf. [52, 55]. Surprisingly, all typical loudspeaker-based reproduction still exhibits pronounced shortcomings when used to synthesize extended perceptually diffuse, and therefore consistently non-directional, sound fields:

Object-based rendering using amplitude panning, channel-based rendering, or scene-based Ambisonic rendering on moderately dense loudspeaker layouts can be interpreted to exhibit a notoriously small sweet area regarding their capacity of rendering uniformly diffuse listener envelopment. Recent studies revealed that instead of the conventional point-source loudspeakers, vertical line-source loudspeakers should be able to supply a larger sweet area with a perceptually diffuse sound field [78–81]. With conventional point-source loudspeakers, the directional impression would often be dominated by the loudspeakers closest to the off-center listener location, even if the off-center displacement is limited to, say, a third of the layout radius [80].

A lot of research drive behind WFS was fueled by the idea that its many loudspeakers and well-controlled time delays avoid a confined sweet area by shaping consistent and extended wave fronts [60]. Plane waves were thought of as an ideal virtual source type to consistently supply a large listening area. And yet, experiments using 2.5D WFS with uncorrelated virtual plane waves showed that the result was only perceived as non-directional when the 2.5D reference line correcting the synthesized plane-wave levels was specific to the listening position [82]. So also 2.5D wave-field synthesis appears limited in rendering a perceptually diffuse sound field to an extended audience. WFS theory suggests [49, 50, 52, 54, 83] that vertical line sources loudspeakers are the optimal sound sources for 2D WFS, of which the generated sound field is constant along the vertical dimension.

A systematic theory is desirable, and might contribute to a profoundly better handling and understanding of diffuse sound field synthesis that may serve many application areas, such as spatial audio and technical acoustics.

1.1 Background and motivation

A noteworthy quote by Jacobsen and Roisin describes diffuse sound fields [84] in the common understanding, based on sources at a large distance:

[…M]ost acousticians would agree on a definition that involves sound coming from all directions. This leads to the concept of a sound field in an unbounded medium generated by distant, uncorrelated sources of random noise evenly distributed over all directions. Since the sources are uncorrelated there would be no interference phenomena in such a sound field, and the field would therefore be completely homogeneous and isotropic. For example, the sound pressure level would be the same at all positions, and temporal correlation functions between linear quantities measured at two points would depend only on the distance between the two points. The time-averaged sound intensity would be zero at all positions. An approximation to this ‘‘perfectly diffuse sound field’’ might be generated by a number of loudspeakers driven with uncorrelated noise in a large anechoic room […]

These properties summarize the common desires and ideals, and for diffuse sound field synthesis with loudspeakers in the free field, it would be interesting to maximize the size of synthesis that approximates these ideals.

This article is devoted to defining ideal continuously surrounding source layers of uncorrelated sources with uniform strength, whose interior spans a field of finite diameter. In particular, relations to potential theory will be exploited to define such ideal layers and their properties. To motivate the underlying research questions, we first explore the properties of some basic source layouts. As common metrics, cf. [8, 20, 85, 86], we employ potential sound energy density w, active sound intensity I, and diffuseness ψ in the sound field, which are described by the sound pressure p and sound particle velocity vector v,

(1)

(2)

with the density of air , the speed of sound , and the statistical expectation E{⋅}. In a perfectly diffuse sound field, these quantities are expected to be

(3)

Otherwise, diffuseness becomes ψ = 0 for a non-diffuse sound field from a single source.

To observe whether w as a measure of the sound pressure level is constant and the active sound intensity I vanishes, we present a small simulation study motivating the particular research questions of this paper.

Figure 1 shows a simulated free-field map of the sound intensity (vectors), the diffuseness (colors), and the sound energy density (contours) along the horizontal plane, for circular or spherical surfaces S of acoustic sources radiating uncorrelated signals of equal variance from many uniformly distributed directions −u as in [14]. This simplifies the integrals of Section 3.1 to

(4)

and ψ is determined by equation (2). The Green’s function decays with for a line source and |Gl| ∝ 1/rl for a point source, cf. equation (13). Distance rl and normalized direction vector ul are measured from each point or line source to the receiver, and normalizes 2ρc2w(0) = 1. Point sources yield the mapping in Figure 1a when arranged in parallel lines and Figure 1c when arranged in a circle, vertical line sources arranged similarly yield the mappings in Figures 1b and 1d, respectively.

thumbnail Figure 1

Horizontal map of potential sound energy density (contours), normalized active intensity I/(2 c w) (arrows), and diffuseness w with 200+200 point sources (a) or vertical line sources (b) equally spaced along parallel lines of a 6:1 length:distance ratio, 100 point sources (c) or vertical line sources (d) spaced by 3.6° in azimuth for a circle (salmon), and with a sphere of 480 uncorrelated point sources (e) at Chebyshev-type nodes from [87]; all driven by statistically independent signals of uniform variance.

The parallel and circular arrangements of point sources Figures 1a and 1c appear to be incapable of rendering a uniformly high diffuseness, and their active sound intensity will be dominated by the nearest source. Potential sound energy density or sound pressure level also increases near the sources. By contrast, the parallel and circular arrangements Figures 1b and 1d of vertical line sources and the spherical arrangement Figures 1e of point sources yield a perfectly vanishing active intensity (or diffuseness of ψ = 100%) everywhere inside, while potential sound energy density w still slightly increase towards the sources.

These simulated fields motivate our research questions about diffuse sound field synthesis with a surrounding layer of uncorrelated sources at finite distance:

  1. Can we formulate conditions that cause a vanishing active sound intensity everywhere inside?

  2. Can there be isotropy at every observation point?

  3. Is a constant sound pressure level achievable?

  4. Can two-point correlations depend only on spacing?

1.2 Outline

Section 2 reviews the common (spherically) isotropic diffuse field model based on uncorrelated plane waves in D dimensions and proves its above-mentioned properties. Section 3 introduces uncorrelated source distributions at a finite distance, statistical expectations for active sound intensity and potential sound energy density, and Gauß’ divergence theorem that would measure the total enclosed variance of all sources. The expected intensity reveals a relation to the solution of Green’s functions of potential fields (gravity, electrostatics) [88, 89].

Section 4 deals with question 1, and Gauß’ divergence will be used to prove that the expected active sound intensity can be forced to vanish inside ideal hollow source layers, e.g., a sphere or infinite cylinder in Section 4.3.

Expanding on question 1 and touching question 3, Section 4.7 shows the formal equivalence of the problem to Newton’s spherical shell theorem [90, Prop. 70, Sect. 12] extended to a generalized number of space dimensions. It helps to understand the required balance of opposite-direction active intensity contributions that cancel for any direction, at any observer location.

Concerning question 2 on isotropy, Section 5 uses the integration elements of Newton’s spherical shell theorem to investigate whether isotropic sectoral contributions to active sound intensity are obtained at every observer position inside the hull.

To deal with question 3, Section 6 establishes the scalar potential of active intensity because it is a gradient field in the specific application. It must be constant for ideal layers, and comparison to potential sound energy density shows whether it can be constant.

Section 7 investigates question 4 about whether the correlation inside only depends on distance.

2 Isotropic diffuse field model

This section reviews the typical isotropic, diffuse plane-wave sound field model and demonstrates its above-mentioned properties of constant sound pressure level, vanishing time-averaged (active) intensity, and point-to-point-distance dependent correlation, everywhere.

A single plane wave from the direction denoted by the unit vector u contributes to the sound pressure p(x) observed at the position x and to the sound particle velocity v(x) observed there; is the wave number, ω = 2πf contains the frequency f, and the imaginary constant is .

Weighted by an amplitude q(u) and superimposed for all surrounding direction vectors in D dimensions, where denotes the D-dimensional unit sphere, the isotropic diffuse field is modeled by these integrals

(5)

(6)

The above-mentioned properties are easily demonstrated when the amplitudes stem from Gaussian random processes q(u) of isotropic variance E{|q(u)|2} = σ2 that are uncorrelated for any pair of different directions

(7)

The Dirac delta δ(u1 – u2) is zero unless u1 = u2, normalized when integrated over the directions, and possesses the sifting property .

  1. The active intensity vanishes at any position x, using for simplification and equation (7),

(8)

  1. The expected sound pressure level denoted as 2ρc2w = E{|p|2} is constant with regard to the position x, using equation (7) with σ2 = 1,

(9)

which is the area of the unit sphere , , or for the specific numbers of space dimensions D = 1, 2, 3, or in general , cf. [91, vol.2, p.387], with the gamma function Γ(⋅), [92, Ch.5].

  1. Finally, the position x also cancels from the correlation of the sound pressure p1 = p(x), p2 = p(x + Δx) at two points because of equation (7), so that correlation only depends on distance Δx = ∥Δx∥ and becomes as derived for arbitrary dimensions D in Appendix A,

(10)

or as summarized in Table 1 using the specific functions for D = 1, 2, 3. The higher the number of dimensions D gets that are used for the isotropic distribution of plane-wave directions, the faster correlation decays over kΔx. Whereas the one-dimensional case D = 1 may be uninteresting for most applications, it is typical to regard the D = 2, 3 cases, cf. [43].

Table 1

Correlation function C(Δx) of the sound pressure at two points spaced by Δx in the ideally isotropic diffuse sound field of D = 1, 2, 3 space dimensions.

3 Distributed, uncorrelated sources

A sound field p(x) that we aim to generate shall be superimposed from an amplitude distribution q(x) that acts as a source term to excite the Helmholtz equation, cf. [93],

(11)

with the Laplacian denoting the second order Cartesian derivatives for all coordinates xi.

Typically, the inhomogeneous problem is solved via the Green’s function of the free field, cf. [93],

(12)

which is the wave of an excitation at a single source point xs to which r=∥x − xs∥ is the Euclidean distance. This excitation is expressed by the Dirac delta that is zero everywhere else than x = xs, normalized , and sifting . The free-field Green’s function must obey Sommerfeld’s radiation condition [94, §28], see derivations in Appendix B and [94, Ch.V(A.IV,C), E13], using the Hankel function of the second kind [92, 10.2.E5], Table 2. The free-field Green’s function is

(13)

with the amplitude . The sound pressure p(x) is obtained by superimposing Green’s functions G weighted by q(xs) for every source point: Expanding the right-hand side of equation (11) as implies with equation (12) the solution on the left-hand side of equation (11)

(14)

Table 2

Green’s function G(r) of the Helmholtz equation in D = 1, 2, 3 space dimensions.

The sound particle velocity is , and accordingly

(15)

3.1 Expected energy density and intensity

The active intensity is, cf. Section B.4 equation (B9),

(16)

The connecting line between any source point xs and any point of observation x defines a unit-length direction vector .

The potential sound energy density 2ρc2w = E{|p|2} scaled by 22ρc2 equals the expected magnitude-square sound pressure, cf. Section B.5 equation (B12),

(17)

These volume integrals equation (16) and equation (17) will only be reduced later to a thin surface layer in equation (24) of Section 4.2 and equation (33) of Section 6, respectively.

4 Ideal layout for zero active intensity

This section shows how ideal source geometries of uniform variance are able to enforce an active intensity that vanishes inside a volume of observation.

4.1 Gauß’ law and sound power

Sound power Pac is typically defined by integrating the divergence of the active intensity over the source volume, e.g. [85], , and Gauß’ law allows to determine it by alternatively integrating the intensity flux aligned with the unit-length surface normal vector n out of the hull,

(18)

We inspect the integrand that describes the contribution to active intensity in equation (16). One can relate it to the gradient of the Green’s function of the potential equation, see Appendix C and Table 3 or [94, Ch. II.E],

(19)

Table 3

Green’s function of the potential equation for D = 1, 2, 3 space dimensions and their derivative.

Since is normalized by the area of the unit sphere in D dimensions , and the gradient of uses , we find

(20)

which shows that equation (16) yields a gradient field. Divergence of the gradient determines the integrand of equation (18), by which we solve the integral

(21)

Upon insertion, it yields the total sound power of the incoherent sources enclosed in Vs by Gauß’ law equation (18),

(22)

that is well-known for D = 3: .

4.2 Surrounding thin-shell source layer

In this article, we will work with another observed volume V ∉ Vs that is free of sources inside, so that the sound power Pac vanishes inside, cf. Figure 2,

(23)

thumbnail Figure 2

Layout of uncorrelated source shell S.

We moreover reduce the volume source distribution in Vs to a thin shell S ⊂ Vs of sources distributed tightly around the observed volume V, but not contained within yet, Vs, S ⊈ V, so that the interior remains source-free. Reduction of the volume integral turns the active sound intensity equation (16) into a surface integral, using equation (20),

(24)

with a factor for normalization. The surface element dS is enclosed in an infinitesimally small angular cone dΩ around u, intersecting with the surface normal n at an angle ϕ. The surface element scales by the reciprocal of the projection cosϕ = n⊺u and by a distance-related area increase rD−1 so that , cf. Figure 2.

4.3 Ideal thin-shell geometry

The source-free interior of the volume V as in equation (23) can only be exploited under geometry assumptions determining the active intensity in equation (23) and forcing it to zero everywhere in V. If, as in Figure 3, active intensity:

  • has a uniform expected magnitude at every surface position ∥I(xs)∥ = In ∀xs ∈ ∂V and

  • is non-tangential, so aligns with the surface normal at every position I(xs) = Inn(xs) ∀xs ∈ ∂V

thumbnail Figure 3

Rotation-invariant and shift-invariant symmetries enforce intensity aligned with surface normal n.

then it must vanish on ∂V due to the source-free interior (Pac = 0) and a non-zero area ()

(25)

which causes a zero active intensity I ≡ 0 ∀x ∈ V.

Proof: I can be defined as gradient ρcI = −∇UI of a potential UI due to equations (16) and (20). It is source-free in V cf. equation (23) so that △UI = 0 in V, and for the product UI∇UI, Gauß’ divergence theorem/Green’s first identity yields

(26)

where assuming In ≡ 0 on yields , whose proportion to can only hold if also I vanishes uniformly inside V.

The constraints for I(xs) require rotation invariance and shift invariance of the expected amplitude variance σ2(xs). Rotation invariance σ2 (Rxs) = σ2(xs) uses a rotation matrix R. It may be limited to a subspace of dimensions if there is shift invariance σ2(xs) = σ2(xs + Δx) with regard to the other E = D – D‣ dimensions σ2(xs) = σ2(xs + Δx), cf. Figure 3.

The discussion of this section assumes incoherent sources for all the D space dimensions that are occupied.

4.4 Ideal source layer for three dimensions

For D = 3, ideal source coordinates lie at a constant radius and describe with D′ = 3, 2, 1 rotation-invariant plus complementing E = 0, 1, 2 shift-invariant dimensions for σ2:

0: a sphere, ,

1: an infinite cylinder, e.g., ,

2: a pair of opposing infinite planes, e.g., .

This first case relates to Figure 1e that demonstrated the sphere of uncorrelated point sources to yield an ideally vanishing active intensity.

The first and second case explain why a ring of point sources in Figure 1c is not ideal and may produce a non-zero active intensity: an ideal three-dimensional layout involving a horizontal ring either needs to be a sphere or an infinite cylinder of uncorrelated point sources. Similarly the parallel lines of point sources in Figure 1a lack vertical extent and are therefore not ideal.

4.5 Ideal source layer for two dimensions

In D = 2space dimensions, ideal source layers with constant radius may use D′ = 2,1 rotation-invariant dimensions when complemented with E = 0, 1 shift-invariant dimensions for σ2:

0: a circle, ,

1: a pair of opposing infinite lines, e.g., .

While cases for fewer than three dimensions may seem practically irrelvant, they can be quite useful, in fact. The first case demonstrates that the circle of uncorrelated vertical line sources in Figure 1d is an ideal layout: Its linear sound field quantities are independent of the vertical z coordinate, therefore imply vertically coherent line sources. Similarly, the second case of opposing parallel lines confirms the ideal condition in Figure 1b.

4.6 Ideal source layer for one dimension

In D = 1 space dimensions, only the E = 0 case remains with flip invariance for σ2:

0: a pair of opposing points, e.g., .

While this case appears to be the least valuable one, by only using two opposing plane waves that are uncorrelated, it still holds valuable insights and is perfectly 1D-isotropic: It yields a vanishing active intensity, a uniform sound pressure level, and its correlation C = cos(kΔx) is ideal by only depending on kΔx, cf. Table 1.

4.7 Newton’s spherical shell theorem

Newton’s original spherical shell theorem [90, Prop. 70, Sect. 12] proves that the gravitational force caused by a thin hollow spherical shell of mass vanishes at every position x enclosed. While essentially equivalent to the symmetry assumptions above, it is geometrically insightful. The contribution dF of a surface element dS at xs to the force at x is proportional to the factor (inverse square law for D = 3 of ) of its distance r = ∥x − xs∥ times , related to the angular element dΩ, as described above. The force contributions dF−- and dF+ of opposing intersections along the directions ±u of any straight line annihilate

(27)

whenever both intersection angles match ϕ+ = ϕ−. For any x, this the case for any opposing pair of intersection points and on a hyper-spherical manifold, by the isosceles triangle enclosed with the manifold’s origin xi = 0 ∀ i ≤ D′, and more obviously for parallel layers, cf. Figure 4 (middle, bottom). With this match for any ±u, integration over all directions u yields a vanishing net force F = 0. Despite a global sign difference, dI = −dF, the formulation for intensity is equivalent.

thumbnail Figure 4

In Newton’s spherical shell theorem, intersection angles only match ϕ+ = ϕ− under rotation (middle) or shift (bottom) invariance, not in general (top).

Assuming symmetry ϕ+ = ϕ− and the closer side of the hull along u at the distance r+ and the longer distance r− to the opposite side, Newton’s theorem shows that a modified source decay 1/r2β may cause imbalance: Contributions and are exactly balanced with the decay 2β = D − 1. Larger exponents cause a dominance of the closest side |dI+|, smaller ones of the more distant one |dI−|.

5 Why isotropy may fail off-center

Isotropy was expressed as the uniformity of the magnitude for all plane-wave directions received [19], [6, here: Richtungsdiffusität], or their spherical wave-number spectra [22]. Equivalently, we may assess isotropy via the sectoral active intensity from an infinitesimal angular cone dΩ aligned with a variable direction u, corresponding to an equation side of equation (27), but with arbitrary σ2,

(28)

so that isotropy can be measured by how constant

(29)

evolves across directions u. For σ2 =1, isotropy is trivial in the isotropic model Section 2 with D = D′, Rs → ∞ or for D = 1, since the direction cosine is unity cosϕ ≡ 1. Otherwise, isotropy is a stricter constraint than Newton’s spherical shell theorem that just requires for every direction u the annihilation with the opposing direction −u by . The strictness of isotropy lies in direction-independence, which is only accomplished (i) at the center of a perfectly spherical layout, D ′= D, σ2 =1, r = Rs = const, and cosϕ ≡ 1, or (ii) at a particular point of observation within an arbitrary convex layout with a source variance

(30)

for every direction u. Assuming to accomplish perfect isotropy by this fixed σ2(us) that matches the respective direction cosine cosϕ0 at every source to a central observer at x = 0, an observer shifted off center will yield a different direction cosine cosϕ at each source, in general, contradicting the assumption. No fixed choice of σ2 is therefore able to enforce isotropy everywhere inside V, neither in an arbitrary convex nor spherical case (ii).

For uniform variance, σ2 = 1, Figure 5 shows the sectoral intensity for the circular/spherical and parallel cases with D = 2,3 with regard to a radial shift x = {0, 0.71, 0.87} of the observer. For the parallels, the peak at ϕ = ϑ = 90° is infinite and position-independent as the direction cosine cosϕ in the denominator always vanishes there; its shift-invariant infinite dimensions dominate sectoral intensity. In the circular/spherical case, sectoral intensity peaks at ϑ = 90°. It is found by and the law of sines of a triangle with sides and opposite angles: shift x and surface angle ϕ, radius Rs and polar angle ϑ.

thumbnail Figure 5

Sectoral intensity to assess isotropy for the 2D circle/3D sphere case (solid) and the 2D parallel line and 3D parallel plane cases (dashed) across observation angle enclosed with the axis of a radial shift of 0% (green), 71% (orange), 87% (blue) of the observer.

6 Why level may not be constant

Because it accomplishes the synthesis of vanishing active sound intensity, we assume a thin source shell in the space and fix the variance to σ2 = 1 at some radius, e.g. Rs = 1 of the unit sphere, denoted as . While both equation (25) and Newton’s spherical shell theorem equation (27) were proof that (hyper-) spherical shells are an ideal geometry for vanishing active intensity,

(31)

this section shows why this may not yield an ideally constant potential sound-energy density w, equation (2), or sound pressure level E{|p|2}, as displayed in Figures 1b–1e in terms of a slight level increase towards the sources.

The surface element is of an isotropic angular segment dΩ0 seen from the origin, the area of the unit sphere is for one less space dimensions (as above: , cf. [91, vol.2, p.387]), the normalizer is , and the subscript D is used to track the number of dimensions.

To utilize the formalism from above for the potential sound energy density w, we employ high-frequency or far-field approximation . It permits a similar discussion as for active intensity, by relating the Green’s functions of the Helmholtz and potential equations everywhere for D = 1, 3 otherwise for large kr,

(32)

cf. Section B.5. For D ≠ 1, 3 the approximation is asymptotic for high frequencies or large distances satisfying . For D = 2, where , a distance to the closest source as small as is approximated well for frequencies above . With equation (32), reduction of the volume integral equation (17) to a hyper-spherical shell integral yields

(33)

It may only be constant for D = 1, as a trivial case with , cf. Table 3. We use the formal similarity to equation (31) to show why.

Since the gradient field equation (31) of active intensity is not yet a scalar counterpart to equation (33), we make use of the potential UI to which it relates with −∇UI = ρcI and which must be constant inside V because I = 0,

(34)

The function involved in the potential UI of equation (34) differs from involved in 2ρc2w of equation (33). We know of to provide a constant UI . Analogy to the discussion in Section 4.7 shows that potential sound energy density 2ρc2w won’t be constant: Its exponent for 1/r is too high, which yields an imbalance towards the closest, dominating sources of the surrounding layer (with the exception of D = 1).

Numerical results with settings as in Section 7 show this increase in Figure 6 for the ideal layers with the 1D case as an exception, and the amount of the level increase reduces by the number of shift-invariant dimensions.

thumbnail Figure 6

Outwards increase of the sound pressure level for the ideal layers (1D: opposing sources, 2D: parallel lines, circle, 3D: opposing planes, cylinder, sphere).

7 Why correlation may be anisotropic

Correlation of the sound pressure at two points and is affected by interference, unlike the quadratic superposition of I and w,

(35)

where . Using and for high frequency and small shift yields , so that

(36)

This equation converges to ideal isotropy of Section 2 equation (10) only if D = D′ and Rs → ∞ so that cosϕ → 1.

We evaluate equation (35) for ideal layouts, σ2 = 1, Rs = 1, at a center x = 0 and an off-center radial position x = [0.87, 0,0]⊺, for radial , tangential , axial shifts separated by Δx = 0.1, with Δs ≈ 0.01 to discretize the integrals. Parallel lines in 2D, Figure 7a, yield a tangential correlation between cos(kΔx) and J0(kΔx), nearly position-independently. Radial correlation is shallower, not as regular, and position-dependent; numerical integration summed up 2000 × 2 sources on 2 lines within ys ∈ [−10; 10] at xs = ±1. For the circle in 2D, Figure 7b, correlation approximates J0(kΔx) best at x = 0, and less so at x = 0.87, where radial and tangential correlations differ; numerical integration summed up 630 sources along a unit circle . Parallel planes in 3D, Figure 7c, yield a tangential correlation between J0(kΔx) and that is nearly position-independent, whereas radial correlation is position-dependent and not as regular; numerical integration summed up 2000 × 2000 × 2 sources on the intervals ys, zs ∈ [−10; 10], at xs = ±1. For the cylinder in 3D, Figure 7d, axial correlation approximates J0(kΔx) nearly position-independently, while radial and tangential correlations are position-dependent and differ; numerical integration summed up 2000 × 630 sources on the axial interval zs ∈ [−10; 10], along unit circles . For the sphere in 3D, Figure 7e, tangential correlation is nearly position-independent and approximates , and radial correlation is position-dependent; numerical integration summed up 130000 sources arranged in a uniform Chebyshev-type spherical layout [87, 130000 (random)]. Apparently, using a maximum number of rotationally symmetric dimensions D′ = D best approximates isotropic correlation.

thumbnail Figure 7

Numerically integrated correlation C(f) for the ideal 2D layers of (a) parallel lines and (b) a circle, the ideal 3D layers of (c) parallel planes, (d) a cylinder, (e) a sphere, in the center x = 0 and off-center by x = 0.87 with R = 1, r2 = 1, and a ∆x = 0.1 of radial (“r”, dotted), axial (“a”, dash-dot), or tangential (“t”, dashed) orientation; light to dark solid lines display the ideal isotropic correlations for the dimensions D = 1, 2, 3, cf. Table 1.

8 Conclusion

In this article we discussed theoretical requirements and limitations of diffuse sound fields synthesized with uncorrelated sources in the free field. The findings are of practical relevance when attempting to produce extended diffuse fields with loudspeakers, either in the anechoic chamber or in typical spatial audio rendering scenarios with loudspeakers or virtual sources. Results demonstrated are accurate in terms of active sound intensity and isotropy for any number of dimensions, and for sound pressure level (potential sound energy density) they are accurate either for D = 1, 3 dimensions or under far-field conditions .

We could prove there being theoretically ideal source layouts that can synthesize vanishing active sound intensity within the entire field enclosed. This was accomplished by statistical expectation and by assuming a continuous layer of uncorrelated surrounding sources of uniform strength. Hereby, the active intensity of superimposed uncorrelated Green’s functions of the Helmholtz equation assumed the form of superimposing the gradient fields of Green’s functions of the potential equation. Potential theory permitted interpretations accessible through Gauß’ divergence theorem and Newton’s spherical shell theorem. Here is a list of ideal layouts:

  • A uniform spherical layer of uncorrelated point sources is able to produce vanishing sound intensity in 3D, i.e. 100% diffuseness, everywhere inside.

  • A circular layer of uncorrelated sources produces 100% diffuseness everywhere inside in 2D; an infinite cylindrical source layer is needed in 3D.

  • Opposing uncorrelated sources yield 100% diffuseness between them in 1D; infinite parallel layers of sources are needed in 2D (lines) and 3D (planes).

We could prove for ideal source layers that sound pressure level is not ideally constant in the entire field enclosed and slightly increases towards the source layer, unless plane-wave sources are used.

We could prove that the ideal, uncorrelated 2D and 3D source layers cannot provide ideal isotropy at every observer position inside. Direction cosines cosϕ to the surface normal cause anisotropy off-center for circular 2D or spherical 3D layouts, or cause a dominance of the shift-invariant dimensions everywhere.

We could moreover prove for the field enclosed by ideal source layers that correlation measured between a pair of points depends not only on distance, but also on the orientation and position, in particular when there is high anisotropy.

Future work should try to solve the integrals involved analytically, find optimal source variances for less ideal geometries (e.g. elliptic, cuboid, and hemi-spherical layouts) and investigate the diffuseness limits of discrete instead of continuous source layers. Moreover, properties of 2.5D WFS as a method to synthesize virtual source arrangements should be investigated. Modified sources should be used to explore options for the non-ideal sound pressure level, isotropy, and correlation metrics.

Acknowledgments

We gratefully received funding from the Austrian Science Fund (FWF): P 35254-N (Envelopment in Immersive Sound Reinforcement, EnImSo). We thank the editor and anonymous reviewers for their appreciation of the topic and guidance through substantial revisions.

Conflict of interest

The authors declare no conflict of interest

Data availability statement

The research data associated with this article are available in the iem git repository, under the reference [95].

Appendix A

Correlation in the diffuse field

The correlation in the D-dimensional, ideally isotropic diffuse sound field uses the integral presented in equation (10) of Section 2 yields

(A1)

which was simplified by the rotationally symmetric integration element , where z = cosϑ is the direction cosine with regard to the axis of the shift Δx, and the area of the unit sphere in D dimensions is . Using the integral from [92, 10.9.E4], the equation becomes in general, for D dimensions,

(A2)

For the particular cases of D = 1,3 dimensions, integration is rather straight-forward, D = 2 uses the typical integral definition [92, 10.9.E1] of the Bessel function,

(A3)

Appendix B

Helmholtz Green’s function

Knowing that the free-field Green’s function only depends on , we may define it via the Laplacian in terms of the radius r, only . The corresponding homogeneous equation simplifies by multiplication with r2 and substitution x = kr to

(B1)

According to Sommerfeld [94, Ch.V(A.IV,C)], transform via the product G = ax−νv(x) of a suitable power ν of x, a scalar a, and a solution v(x) is useful and leaves after inserting and dividing by ax−ν

(B2)

B.1 Far-field or D = 1,3 solution

The specific choice 2ν = D − 1 turns equation (41) into the differential equation of an undamped oscillator, v = e±ix), whenever ,

(B3)

which yields . This is either precise for ν = 0, 1, so D = 1, 3, or it is a far-field approximation for D ≠ 1, 3, otherwise.

B.2 General solution for any

The standard substitution 2ν = D − 2 for a general solution turns equation (B2) into the Bessel differential equation

As a possible solution for v, the Hankel function of the second kind [92, 10.2.E6] yields with G = ax−νv,

(B4)

This is the physical solution that fulfills Sommerfeld’s radiation condition [94, §28].

B.3 Scaling for any D

While in equation (B4) with x = kr captures how the Green’s function evolves over any radius r > 0 as homogeneous solution, a suitable scaling a(k) must be found in equation (B4) so that G matches its inhomogeneity −δ(x) in equation (12). This is done by integrating equation (12) over a spherical volume

(B5)

whose radius r we are allowed to minimize by without changing , so that we can utilize as a simplification the small-radius approximation of the Hankel function , cf. [92, 10.7.E7] as in [94, Ch.V(A.IV,C)]. For G and its derivative , it yields:

First, substitution and 2ν = D − 2 is used to show that vanishes in equation (B5)

Then what remains is integrated using the divergence theorem ,

(B6)

With the small-radius form from above and , this yields a:

Together with x−vHν(x), it is used in Section 3 equation (13)

(B7)

B.4 Velocity and active intensity for any D

The radial velocity of equation (B7) is, using [92, 10.6.E1],

(B8)

Multiplied with , it yields the active radial intensity of the single source

(B9)

which was simplified by the Wronskian [92, 10.5.E5] and yields the expression in equation (16) of Section 3.1.

B.5 Far-field potential sound-energy density for any

The potential sound energy density 2ρc2w = |p|2 becomes with equations (B7) and (B8) for the single source,

and with the far-field approximation [92, 10.2.E6]

(B10)

the far-field Green’s function becomes

(B11)

and we obtain for the potential sound energy density

(B12)

These equations are applied in Section 3 equation (17) and Section 6 equation (32), and their common amplitude term is

(B13)

Appendix C

Potential Green’s function

The Green’s function of the potential equation fulfills

(C1)

and for , we have a homogeneous equation

(C2)

that is fulfilled by the power

(C3)

and using Gauß law, as before, we can find the constant a of the inhomogeneous problem for any r > 0

(C4)

as used in Section 4 equation (19) and Table 3.

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Cite this article as: Zotter F. Riedel S. Gölles L. & Frank M. 2024. Diffuse sound field synthesis: Ideal source layers. Acta Acustica, 8, 34.

All Tables

Table 1

Correlation function C(Δx) of the sound pressure at two points spaced by Δx in the ideally isotropic diffuse sound field of D = 1, 2, 3 space dimensions.

In the text
Table 2

Green’s function G(r) of the Helmholtz equation in D = 1, 2, 3 space dimensions.

In the text
Table 3

Green’s function of the potential equation for D = 1, 2, 3 space dimensions and their derivative.

In the text

All Figures

thumbnail Figure 1

Horizontal map of potential sound energy density (contours), normalized active intensity I/(2 c w) (arrows), and diffuseness w with 200+200 point sources (a) or vertical line sources (b) equally spaced along parallel lines of a 6:1 length:distance ratio, 100 point sources (c) or vertical line sources (d) spaced by 3.6° in azimuth for a circle (salmon), and with a sphere of 480 uncorrelated point sources (e) at Chebyshev-type nodes from [87]; all driven by statistically independent signals of uniform variance.
In the text
thumbnail Figure 2

Layout of uncorrelated source shell S.
In the text
thumbnail Figure 3

Rotation-invariant and shift-invariant symmetries enforce intensity aligned with surface normal n.
In the text
thumbnail Figure 4

In Newton’s spherical shell theorem, intersection angles only match ϕ+ = ϕ− under rotation (middle) or shift (bottom) invariance, not in general (top).
In the text
thumbnail Figure 5

Sectoral intensity to assess isotropy for the 2D circle/3D sphere case (solid) and the 2D parallel line and 3D parallel plane cases (dashed) across observation angle enclosed with the axis of a radial shift of 0% (green), 71% (orange), 87% (blue) of the observer.
In the text
thumbnail Figure 6

Outwards increase of the sound pressure level for the ideal layers (1D: opposing sources, 2D: parallel lines, circle, 3D: opposing planes, cylinder, sphere).
In the text
thumbnail Figure 7

Numerically integrated correlation C(f) for the ideal 2D layers of (a) parallel lines and (b) a circle, the ideal 3D layers of (c) parallel planes, (d) a cylinder, (e) a sphere, in the center x = 0 and off-center by x = 0.87 with R = 1, r2 = 1, and a ∆x = 0.1 of radial (“r”, dotted), axial (“a”, dash-dot), or tangential (“t”, dashed) orientation; light to dark solid lines display the ideal isotropic correlations for the dimensions D = 1, 2, 3, cf. Table 1.
In the text

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